21.
If
4
5
g t dt
, then
4
2
6
g t
dt
7
7
(A)
15
(B)
16
(C)
22
(D)
23
(E)
28
AP Calculus Review
6:
Area and Volume
1.
The function
f
is continuous on the interval [4, 16].
The values of
f
at selected values of
x
are shown in the
table below.
Use a trapezoid rule to estimate the area under the graph of
f
on the interval [4, 16].
x
4
6
9
13
16
f
x
8
11
15
12
10
2.
Find the area bounded by the graph of
2
20
6
10
f
x
x
x
and the line
y
= 4.
3.
Let the region
R
in Quadrant I be bounded by the graphs of
3
1/3
y
k x
and
2
2
x
y
k
where
k
is a positive
constant.
a.
Find the area of
R
b.
Find the rate of change of the area of
R
with respect to
c. Suppose
k
is decreasing at a rate of 0.2.
Find the rate of change of the area of
R
when
k
d.
Let
k
= 2.
Find the value of
b
such that the line
y
=
b
divides the region
R
into 2 equal pieces.
4. Let
R
be the region in the first quadrant bounded by the graph of
y
= 8 –
x
3
and the coordinate axes.
a.
Find the volume that results when
R
is rotated about the
x
axis.
b. Find the volume that results when
R
is rotated about the
y
axis.
c.
Find the volume that results when
R
is rotated about the line
y
= 8.
d. Let
R
be the base of a solid.
Each cross section perpendicular to the
x
axis is a rectangle with its base in
the
x

y
plane and its height given by
h
(
t
) =
x
2
.
Find the volume of this solid.
5.
Let the first quadrant region enclosed by the graph of
1
y
x
and the lines
x
= 1 and
x
= 4 be the base of a
solid.
If cross sections perpendicular to the
x
axis are semicircles, the volume of the solid is
.
k
.
= 2.
3
4
6.
The volume obtained when the region in the first quadrant bounded by the
x
axis and the curve
y
= 2
x
–
is revolved about the
x
axis is
x
2
8
7.
The region in the first quadrant bounded by the graph of
y
= arcsin
x
,
y
=
/2 and the
y
axis is rotated
around the yaxis. The volume of the resulting solid is given by
2
2
2
2
2
8.
Let the base of a solid be the first quadrant region enclosed by the
x
axis, the
y
axis and the graph of
2
1
4
x
y
.
If all cross sections perpendicular to the
y
axis are squares, then the volume of the solid is
(A)
3
(B)
2
(C)
1
(D)
1/2
(E)
1/3
AP Calculus Review
7:
Calculus Applications
1.
Find the average value of
3
2
8
12
f
x
x
x
x
on the interval in the first quadrant where
0
f
x
2.
Find the value of
k
such that the average value of
f
x
x
on the interval [0,
k
] is 2.
3.
A particle moves along the
x
axis with a velocity given by
2
sin
12
t
v t
t
for 0 ≤
t
≤ 6.
At time
t
= 0,
the particle was at
x
=
a.
What was the particle’s distance traveled from
t
= 0 to
t
= 6 ?
b. What was the particle’s position at
t
= 3 ?
.
2.